In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a 2-group, or 2-dimensional higher group, is a certain combination of
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
and
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*''Group'' with a partial functi ...
. The 2-groups are part of a larger hierarchy of
''n''-groups. In some of the literature, 2-groups are also called gr-categories or groupal groupoids.
Definition
A 2-group is a
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...
''G'' in which every morphism is invertible and every object has a weak inverse. (Here, a ''weak inverse'' of an object ''x'' is an object ''y'' such that ''xy'' and ''yx'' are both isomorphic to the unit object.)
Strict 2-groups
Much of the literature focuses on ''strict 2-groups''. A strict 2-group is a ''strict'' monoidal category in which every morphism is invertible and every object has a strict inverse (so that ''xy'' and ''yx'' are actually equal to the unit object).
A strict 2-group is a
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
in a
category of categories
In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-c ...
; as such, they are also called ''groupal categories''. Conversely, a strict 2-group is a
category object in the
category of groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
Relation to other categories
There a ...
; as such, they are also called ''categorical groups''. They can also be identified with
crossed module In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H by automorphisms (which we will write on the left, (g,h) \mapsto g \cdot h , and a homomorphism of groups
: d\colon H \longrightarro ...
s, and are most often studied in that form. Thus, 2-groups in general can be seen as a weakening of
crossed modules In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H by automorphisms (which we will write on the left, (g,h) \mapsto g \cdot h , and a homomorphism of groups
: d\colon H \longrightar ...
.
Every 2-group is equivalent to a strict 2-group, although this can't be done coherently: it doesn't extend to 2-group homomorphisms.
Properties
Weak inverses can always be assigned coherently: one can define a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
on any 2-group ''G'' that assigns a weak inverse to each object and makes that object an
adjoint equivalence in the monoidal category ''G''.
Given a
bicategory
In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative ''up to'' an isomor ...
''B'' and an object ''x'' of ''B'', there is an ''automorphism 2-group'' of ''x'' in ''B'', written Aut
''B''(''x''). The objects are the
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s of ''x'', with multiplication given by composition, and the morphisms are the invertible 2-morphisms between these. If ''B'' is a
2-groupoid (so all objects and morphisms are weakly invertible) and ''x'' is its only object, then Aut
''B''(''x'') is the only data left in ''B''. Thus, 2-groups may be identified with one-object 2-groupoids, much as groups may be identified with one-object groupoids and monoidal categories may be identified with one-object bicategories.
If ''G'' is a strict 2-group, then the objects of ''G'' form a group, called the ''underlying group'' of ''G'' and written ''G''
0. This will not work for arbitrary 2-groups; however, if one identifies isomorphic objects, then the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es form a group, called the ''fundamental group'' of ''G'' and written π
1(''G''). (Note that even for a strict 2-group, the fundamental group will only be a
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
of the underlying group.)
As a monoidal category, any 2-group ''G'' has a unit object ''I''
''G''. The
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of ''I''
''G'' is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
by the
Eckmann–Hilton argument
In mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument about two unital magma structures on a Set (mathematics), set where one is a homomorphism for the other. Given this, the stru ...
, written Aut(''I''
''G'') or π
2(''G'').
The fundamental group of ''G''
acts on either side of π
2(''G''), and the associator of ''G'' (as a monoidal category) defines an element of the
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
H
3(π
1(''G''),π
2(''G'')). In fact, 2-groups are
classified in this way: given a group π
1, an abelian group π
2, a group action of π
1 on π
2, and an element of H
3(π
1,π
2), there is a unique (
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
equivalence) 2-group ''G'' with π
1(''G'') isomorphic to π
1, π
2(''G'') isomorphic to π
2, and the other data corresponding.
The element of H
3(π
1,π
2) associated to a 2-group is sometimes called its Sinh invariant, as it was developed by
Grothendieck's student
Hoàng Xuân Sính
Hoàng Xuân Sính (born September 8, 1933) is a Vietnamese mathematician, a student of Grothendieck, the first female mathematician in Vietnam, the founder of Thang Long University, and the recipient of the ''Ordre des Palmes Académiques''.
Ea ...
.
Fundamental 2-group
Given a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
''X'' and a point ''x'' in that space, there is a fundamental 2-group of ''X'' at ''x'', written Π
2(''X'',''x''). As a monoidal category, the objects are
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
s at ''x'', with multiplication given by concatenation, and the morphisms are basepoint-preserving
homotopies between loops, with these morphisms identified if they are themselves homotopic.
Conversely, given any 2-group ''G'', one can find a unique (
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
weak homotopy equivalence In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category.
A model category is a category with cla ...
)
pointed connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties tha ...
''(X,x)'' whose fundamental 2-group is ''G'' and whose
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s π
''n'' are trivial for ''n'' > 2. In this way, 2-groups
classify pointed connected weak homotopy 2-types. This is a generalisation of the construction of
Eilenberg–Mac Lane spaces.
If ''X'' is a topological space with basepoint ''x'', then the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of ''X'' at ''x'' is the same as the fundamental group of the fundamental 2-group of ''X'' at ''x''; that is,
:
This fact is the origin of the term "fundamental" in both of its 2-group instances.
Similarly,
:
Thus, both the first and second
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s of a space are contained within its fundamental 2-group. As this 2-group also defines an action of π
1(''X'',''x'') on π
2(''X'',''x'') and an element of the cohomology group H
3(π
1(''X'',''x''),π
2(''X'',''x'')), this is precisely the data needed to form the
Postnikov tower
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with the ...
of ''X'' if ''X'' is a pointed connected homotopy 2-type.
See also
*
N-group (category theory)
In mathematics, an ''n''-group, or ''n''-dimensional higher group, is a special kind of ''n''-category that generalises the concept of group to higher-dimensional algebra. Here, n may be any natural number or infinity. The thesis of Alexander G ...
*
Abelian 2-group
In mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and ...
References
*
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External links
*
* 200
Workshop on Categorical Groupsat the
Centre de Recerca Matemàtica
The Centre de Recerca Matemàtica (CRM) () is a consortium, with its own legal status, integrated by the Institut d'Estudis Catalans (IEC) and the Catalan Government. It is a research institute associated with the Universitat Autònoma de Barcelon ...
{{Category theory
Group theory
Higher category theory
Homotopy theory